Exclusive semileptonic $B_s\to K \ell \nu$ decays on the lattice

Semileptonic <math display="inline"><msub><mi>B</mi><mi>s</mi></msub><mo stretchy="false">→</mo><mi>K</mi><mo>ℓ</mo><mi>ν</mi></math> decays provide an alternative <math display="inline"><mi>b</mi></math>-decay channel to determine the Cabibbo-Kobayashi-Maskawa (CKM) matrix element <math display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo></math> and to obtain a <math display="inline"><mi>R</mi></math>-ratio to investigate lepton-flavor-universality violations. Results for the CKM matrix element may also shed light on the discrepancies seen between analyses of inclusive or exclusive decays. We calculate the decay form factors using lattice QCD with domain-wall light quarks and a relativistic <math display="inline"><mi>b</mi></math>-quark. We analyze data at three lattice spacings with unitary pion masses down to 268 MeV. Our numerical results are interpolated/extrapolated to physical quark masses and to the continuum to obtain the vector and scalar form factors <math display="inline"><msub><mi>f</mi><mo>+</mo></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> and <math display="inline"><msub><mi>f</mi><mn>0</mn></msub><mo stretchy="false">(</mo><msup><mi>q</mi><mn>2</mn></msup><mo stretchy="false">)</mo></math> with full error budgets at <math display="inline"><msup><mi>q</mi><mn>2</mn></msup></math> values spanning the range accessible in our simulations. We provide a possible explanation of tensions found between results for the form factor from different lattice collaborations. Model- and truncation-independent <math display="inline"><mi>z</mi></math>-parametrization fits following a recently proposed Bayesian-inference approach extend our results to the entire allowed kinematic range. Our results can be combined with experimental measurements of <math display="inline"><msub><mi>B</mi><mi>s</mi></msub><mo stretchy="false">→</mo><msub><mi>D</mi><mi>s</mi></msub></math> and <math display="inline"><msub><mi>B</mi><mi>s</mi></msub><mo stretchy="false">→</mo><mi>K</mi></math> semileptonic decays to determine <math display="inline"><mo stretchy="false">|</mo><msub><mi>V</mi><mrow><mi>u</mi><mi>b</mi></mrow></msub><mo stretchy="false">|</mo><mo>=</mo><mn>3.8</mn><mo stretchy="false">(</mo><mn>6</mn><mo stretchy="false">)</mo><mo>×</mo><msup><mn>10</mn><mrow><mo>-</mo><mn>3</mn></mrow></msup></math>. The error is currently dominated by experiment. We compute differential branching fractions and two types of <math display="inline"><mi>R</mi></math> ratios, the one commonly used as well as a variant better suited to test lepton-flavor universality.